Conventional broadband time-domain beam formersfor antenna-array processing require one tapped de-lay line ~TDL! for each antenna element to avoidbeam squint, which is the undesired rotation of theangular receptivity pattern of the antenna with fre-quency. Often these TDL’s are implemented withdigital delay lines for narrow-band systems ~up to afew megahertz of bandwidth!. To process signalswith a bandwidth up to ;1 GHz, TDL’s can be im-
G. Kriehn [email protected]!, A. Kiruluta, P. E. X.ilveira, S. Weaver, S. Kraut, and K. Wagner are with the Optoelec-ronic Computing Systems Center, Department of Electrical andomputer Engineering, University of Colorado, Boulder, Colorado0309-0425. R. T. Weverka is with Optoelectronic Data Systems,32 Hacienda Street, San Mateo, California 94403. L. Griffiths isith the Engineering Department, George Mason University, Fair-
ax, Virginia 22030.Received 24 May 1999; revised manuscript received 29 October
212 APPLIED OPTICS y Vol. 39, No. 2 y 10 January 2000
plemented with ultrasonic delay lines, which can beconveniently tapped by acousto-optic ~AO! diffrac-tion.1 At very high frequencies ~perhaps even asgreat as 100 GHz! fiber-optic TDL’s have beenproposed.2–4 Large arrays can have as many as1000 elements for one-dimensional ~1-D! arrays andas many as 100 3 100 elements for two-dimensional~2-D! arrays, requiring N 5 103–104 broadbandTDL’s, which is both expensive and technologicallychallenging. These TDL’s are necessary for avoid-ing beam squint, which is especially troublesome inlarge arrays for large fractional bandwidth applica-tions. To obtain squint-free octave-bandwidth beamforming over a half-space for a half-wavelengthspaced array, each TDL needs at the minimum asmany complex taps as there are elements in a 1-Dphased array or elements along the longest diagonalfor a 2-D array. Even more taps are required, how-ever, for the equalization of antenna frequency re-sponse ripples, compensation for mutual coupling, ormultipath processing.
To avoid the necessity for a large number of TDL’s,many phased arrays are operated in a narrow-band
aprt
e1Gdfi
tfalcr
aopwe
mode in which each antenna output is simply multi-plied by a single complex coefficient for beam form-ing, as shown in Fig. 1~a!. However, when thefractional bandwidth F 5 Byf0 ~where B is the band-width and f0 is the center frequency! of a signalexceeds the spatial-array resolvability Q 5 l0yLsin umax ~where L is the maximum array aperture,which equals L 5 Nd for an N-element 1-D array ofelement spacing d, l0 5 cyf0, and umax is the maxi-mum angle from boresight over which the array muststeer!, then a plane-wave pulse arriving at a largengle u from boresight takes a time T 5 Lyc sin u toropagate across the array aperture, and this delay isesolvable within the bandwidth of the system. Inhis case, when F . Q, the antenna-array function is
significantly altered as the frequency changes acrossthe bandwidth B, resulting in an undesired angularrotation of the main beam and an additional unde-sired rotation of the nulls of the antenna pattern.
In current implementations of antenna beam form-ers when broadband operation is required ~the con-dition in which F . Q!, a weighted TDL with at leastM 5 sin umax BLyc taps is required at every antennalement, as shown in Fig. 1~b!. As an example, for a-D array with N 5 1000, a center frequency of 2Hz, a processing bandwidth of 1 GHz ~F 5 1y2!, aesired angular coverage from boresight to both end-re directions ~umax 5 90°!, with elements that are
spaced at lmy2 ~where lm is the minimum detectablewavelength of the array!, M 5 200 complex taps arerequired per array element. This results in a re-quirement of NM 5 200,000 complex taps for theentire 1-D array. For a 100 3 100 2-D array oper-ating at the same frequencies, 28 complex taps arerequired per antenna element, and a total of 280,000complex taps are needed for beam forming withoutsquint. In the alternative frequency-domain imple-mentation, a temporal Fourier transform of each an-tenna output is implemented digitally with an arrayof N fast Fourier transform ~FFT! modules and usedto produce M 5 BLyc narrow-band systems, each of
which is steered with a straightforward complex mul-tiplier operating at every antenna element at eachresolvable frequency. After narrow-band beamforming is performed in each frequency bin, the de-sired broadband signal is reconstructed by an inverseFFT of the summed narrow-band outputs. How-ever, the frequency-domain implementation ignorescorrelations between frequency components ~i.e., itassumes an on-diagonal block Toeplitz correlationstructure! that can produce erroneous results in cer-ain broadband situations. In addition, thisrequency-domain system has as many FFT moduless conventional time-domain beam formers have de-ay lines ~N!, and the FFT modules are far moreomplex to implement than a delay line, especially foreal-time operation.
In this paper we describe a novel, more efficientlgorithm for broadband time-domain beam formingf arbitrarily large arrays that requires only one in-ut TDL detector and one output TDL modulator foreight programming. This BEAMTAP @broadbandfficient adaptive method for true-time-delay ~TTD!
array processing# system makes broadband beamforming viable for large arrays and is well matched toimplement with radio frequency ~rf ! photonic hard-ware. In addition, BEAMTAP is compatible withthe real-time calculation of the required MN 5 NFyQadaptive weights that encompass the necessary de-grees of freedom to nearly optimally beam steer andnull rotate without squint in an arbitrarily complexspatiotemporal signal environment.
We begin by comparing the conventional time-domain beam former with the new hardware-efficientBEAMTAP architecture, and we show their mathe-matical equivalence before discussing the benefits andchallenges of the proposed optical array processingsystem as compared with other photonic approachesand digital signal-processing implementations. Thenthe proposed rf photonic implementation with a fiber-remoted coherent phased array, photorefractive crys-tal, traveling-wave detector, and AO device ~AOD! is
Fig. 1. ~a! Narrow-band phased array that suffers from frequency-dependent beam squint. ~b! Conventional broadband time-delay-and-sum beam-forming algorithm illustrating the requirement for one TDL at each antenna element to eliminate beam squint of the mainbeam and the nulls.
10 January 2000 y Vol. 39, No. 2 y APPLIED OPTICS 213
e
dtwncm
OPmfiholBssl
dtHw
tc
tf
2
outlined. The details of the rf offset frequencies, grat-ing recording, coherent interferometric detection, andclosed-loop optical system operation are then derivedin Section 3. Simulations are presented that demon-strate the operations of TTD main-beam forming andjammer nulling for far-field broadband signals. Fi-nally, we conclude with a summary of our results.
2. Optical Phased-Array Radar Processing
There have been numerous approaches to opticallyimplementing adaptive phased-array phase-onlybeam steering, all of which unfortunately suffer frombeam squint for broadband signals.5,6 Previous op-tical implementations of broadband adaptive beamforming7–11 avoided beam squint by use of multichan-nel AO delay lines to implement a TDL for everyantenna element input, but are limited to 32–64channels because of the limits of AO technology.12
Fiber-optic TTD beam-forming networks have beenemployed to bring the main beam onto the apparentarray boresight where there is no frequency-dependent beam squint,2,3,13,14 though squint is stillpresent in the sidelobes and the nulls. Althoughthey have the potential to be scanned rapidly, thesesystems are not adaptive and are able to process onlylinear or planar arrays and not ones that are confor-mal, irregular, or dynamically flexing. In addition,these fiber-based systems do not provide for a mech-anism of adaptively weighting the array function tonull out jammers or simultaneously point multiplebeams. Moreover, fiber-optic TTD networks, eventhe elegant wavelength-tuned dispersive,15 fiberprism,3 or grating-reflective approaches,16 can creatextreme fiber management problems.Our research group under K. Wagner previously
eveloped adaptive phased-array processing sys-ems, using photorefractive crystals as the adaptiveeights that achieved 45 dB of narrow-band jammerulling, and had the potential capability for TTD pro-essing, but in a complex frequency-domain imple-entation.17 A further drawback of our previous
system was that to perform TTD beam forming di-rectly on large arrays in the optical domain it wasnecessary to incorporate as large an optical TDL withas large an optical path length as the size of thephased array—yet over this propagation distance theshort optical wavelength makes diffractive spreadinga significant problem unless reimaging ~as in a stableHerriot cell resonator! or fiber waveguiding is used.
ur previous system used wedged or tilted Fabry–erot etalons to incorporate this delay, which wasanifested as arrays of frequency-selective bandpasslters with a resolution inverse to the required delay;owever, achieving such resolution on parallel arraysf signals presented a significant engineering chal-enge. The photonic implementation of theEAMTAP network presented here is a substantialimplification and outgrowth of this previous re-earch and represents an attractive alternative to theimitations of previous optical approaches.18 The
key is the use of a single traveling-wave detector suchas a 1-D time-delay-and-integrate charge-coupled de-
14 APPLIED OPTICS y Vol. 39, No. 2 y 10 January 2000
vice ~TDI CCD!19,20 or a photoconductive traveling-fringes detector ~TFD!21,22 to allow for the generationof the necessary tap-in time delays in the electrical~rather than the optical! domain, whereas the tap-out
elay line can be implemented compactly with acous-ic delay lines such as AO Bragg cells for the input.owever, all the adaptive timing, phasing, andeighting is still done in the optical domain.
A. Conventional Time-Delay-and-Sum Adaptive BeamFormer
In conventional time-domain beam forming, asshown schematically in Fig. 1 and with the details ofthe least-mean-square ~LMS! algorithm in Fig. 2,each of the N antenna outputs sn~t! ~for n 5 1, . . . , N!is input to a TDL, which is represented for discretedelays as sn~t 2 mt! ~for m 5 0, . . . , M 2 1!, wherethe temporal spacing between taps is t and the totaldelay of the TDL is T [ ~M 2 1!t 5 M9t where M9 ishe total number of time delays and is represented inontinuous notation as sn~t 2 t9! for 0 , t9 , T. The
output consists of a sum of each antenna output caus-ally convolved with finite impulse responses W*nm ofduration T,
o~t! 5 (n51
N
(m50
M21
sn~t 2 mt!W*nm. (1)
The weighted impulse response filter W*nm of eachantenna element is usually calculated by an adaptivealgorithm, and for the common LMS style of algo-rithms23,24 the weights are calculated as a correlationcoefficient between each delayed antenna signal anda signal f ~t!—which is either derived from a feedbackloop or is an a priori known reference signal. Themost common form of error-driven feedback is when
Fig. 2. Conventional time-delay-and-sum approach to adaptivearray processing for broadband squint-free TTD beam formingwith LMS adaptation based on a desired signal and a correlation–cancellation-loop feedback, where the desired signal d~t! is sub-racted from the output signal o~t! to produce the feedback signal~t!. Each weight is formed by integration of the multiplication of
the feedback with the delayed antenna signals. The delayed an-tenna signals are then multiplied by the weights and summed toproduce the output.
I
f
f ~t! 5 d~t! 2 o~t! for some desired signal d~t!. Thusthe weighted impulse response filter is given by
W*nm 5 *2`
t
s*n~t1 2 mt! f ~t1!dt1. (2)
The time dependence of the weights that are due toadaptation up through time t is slow and can often beassumed constant over the duration of the convolu-tion with the signals. Given this form of the adap-tive weight matrix, the output can be written as
o~t! 5 (n51
N
(m50
M21
sn~t 2 mt!W*nm
5 (n51
N
(m50
M21
sn~t 2 mt! *2`
t
s*n~t1 2 mt! f ~t1!dt1
5 (n
(m
sn~t 2 mt! *2`
t
s*n~t1 2 mt!
3 @d~t1! 2 o~t1!#dt1. (3)
Solving for the output o~t! can be accomplished witha frequency-domain approach that yields a power-nulling spatiotemporal filter. Below we show anearly identical output for our new BEAMTAP beamformer that requires only two TDL’s in comparisonwith the N TDL’s required by this LMS time-delay-and-sum conventional approach.
B. BEAMTAP Algorithm
In the new BEAMTAP algorithm shown in Fig. 3,each antenna element output sn~t! is multiplied atevery resolvable instant of time t by a linear array ofweights W*nm, which are located along the corre-sponding row of the weight matrix. Each product issummed along the columns with the correspondingproducts from the other rows produced by the otherarray elements, and the resulting sum is input to atap-in scrolling delay line, whose inputs at the mthtap-in position are given by a weighted sum overundelayed signals
Ym~t! 5 (n51
N
sn~t!W*nm. (4)
As the tap-in delay line spatially scrolls in time byincrements of t, it accumulates the appropriate con-tributions from each column sum position, and aftera particular moving sum traverses the entire delay-line aperture, the resulting output is the desired mul-tiply weighted and delayed sum from the array,
o~t! 5 (m50
M21
Ym~t 2 mt! 5 (m50
M21
(n51
N
sn~t 2 mt!W*nm. (5)
Thus from any antenna element there is a path intothe delay line at a position such that the output has
the appropriate delay, and this is exactly equivalentto the conventional time-domain beam former givenin Eq. ~1!. In a sense we have simply commuted thelinear operation of multiple TDL’s that have oneweighted output from each tap position into a singlemultiple-input-delay line that addresses the same ar-ray of weights.
To operate as an adaptive phased array, theweights must be calculated in response to changes ofthe signal environment. The LMS algorithm can bereadily implemented within the hardware-efficientBEAMTAP beamformer by inclusion of an input de-lay line and a correlator–integrator to calculate eachadaptive weight in the M 3 N array of weights.Each processing element in the weight matrix arraycontains an analog multiplier that multiplies the out-put of the tap-out delay line at that column with thesample from the phased array at that row. Theseproducts are locally time integrated to produce theadaptive weights as the appropriate correlation coef-ficients
W*nm~t! 5 *2`
t
s*n~t1! f $t1 1 @m 2 ~M 2 1!#t%dt1. (6)
n this expression m varies from 0 to M 2 1 so thatthe argument of f at the t limit of integration variesrom t 2 ~M 2 1!t to t and is therefore always causal.
These weights are then used as the coefficients in thesubsequent array of multipliers to produce the appro-priate beam-steered array output as the scrolling
Fig. 3. BEAMTAP algorithm for broadband squint-free TTDbeam forming with a single output tap-in delay line. For adaptivecalculation of the weights within the array, the additional inputTDL is also required. The desired signal d~t! is again subtractedfrom the output signal o~t! to produce the feedback signal f ~t!.Here the variably delayed feedback is multiplied by the antennasignals delayed by T and integrated to produce the weights. Theweights are then multiplied by the undelayed antenna signals,column summed, and input into the tap-in delay line.
10 January 2000 y Vol. 39, No. 2 y APPLIED OPTICS 215
r
a
abmnatwsrawt
S
1
tmsatuoyrtvnds
2
sum of the adaptively weighted and column-summedcontributions:
o~t! 5 (m
Ym~t 2 mt!
5 (m
(n *
2`
t
d@t9 2 ~t 2 mt!#sn~t9!W*nm~t9!dt9
5 (n,m
sn~t 2 mt! *2`
t2mt
s*n~t1! f @t1 1 ~m 2 M9!t#dt1
5 (n,m
sn~t 2 mt! *2`
t
s*n~t2 2 mt! f ~t2 2 M9t!dt2.
(7)
Note that this adaptive algorithm is the exact equiv-alent of the time-domain LMS adaptive beam formershown in Eq. ~3!, except for the simple time shift ofthe correlation signal f~t! by M9t 5 ~M 2 1!t. Forcorrelating against a known reference signal f ~t! asequired in main-beam formation,25,26 this time delay
is readily accommodated by advancing the referenceappropriately. This unwanted delay is more trou-blesome in correlation–cancellation-loop algorithmsas required for jammer nulling, in which the feedbacksignal is given by the desired reference minus theoutput f ~t! 5 d~t! 2 o~t!, since the output cannot beadvanced in time @this would require us to know o~t!before it has been generated#. However, we cansolve this dilemma by delaying the antenna signals,sn~t!, which are used to write the weights relative tothe signals used for reading the weights, by an addi-tional T 5 M9t:
o~t! 5 (n
(m
sn~t 2 mt! *2`
t
s*n~t2 2 mt 2 M9t!
3 f ~t2 2 M9t!dt2
5 (n
(m
sn~t 2 mt! *2`
t2M9t
s*n~t2 2 mt!
3 @d~t2! 2 o~t2!#dt2. (8)
This produces the correct relative delay between sn~t!nd o~t! inside the weight integral, and d~t! can be
delayed as desired. The only difference from Eq. ~3!is the upper limit of the integral. This means thatthe weights are being updated and adapted by thedelayed product of sn~t! and f ~t!, which will be incon-sequential for realistic slow adaptation rates or atsteady-state operation. It should also be noted thatalthough this approach does require a second versionof each antenna signal, sn~t!, delayed by a fixedamount, M9t, it does not require a TDL version ofthese signals.
A note on the convergence properties of this algo-rithm are in order, since conventional LMS arrayprocessing with digital signal-processing techniquesare known to converge only for a limited feedbackgain.27,28 The real-time array adaptation in
16 APPLIED OPTICS y Vol. 39, No. 2 y 10 January 2000
BEAMTAP is a continuous-time adaptive algorithm,since the weights evolve continuously even thoughthe output waveform may be sampled by the quan-tized charge packets of an output discrete shift reg-ister ~the tap-in delay line! in addition to having thearray dimension always being sampled. However,since the adaptive weights are updated continuouslyand are governed by a first-order differential equa-tion that is always guaranteed to converge, increas-ing the feedback gain will not lead to oscillations andunbounded weight growth as in sampled adaptation.Instead, it will simply increase the adaptation speedand increase the null depths.27 It is important tonote, though, that noise in the system and the un-wanted feedthrough of the writing beam onto thevelocity-matched traveling-detector array can lead toloop oscillation and will present a practical limit onthe speed of adaptation.29
The BEAMTAP beam-forming algorithm is espe-cially well suited to a photonic implementation ofadaptive beam-forming and jammer-nulling signalprocessing in very large phased arrays. The opti-cal approach allows for an enormous number ofadaptive weights ~109! to be stored in the volume of
photorefractive crystal, thereby enabling the broad-and processing of very large arrays as the analogultiplication, integration, and final multiplicationeeded to write and read out the weight matrix areccomplished though the interference, grating forma-ion, and subsequent diffraction, respectively, of lightithin the crystal itself. Furthermore, the corre-
ponding digital processing load that would beequired for solving these large, broadband, adaptive-rray signal-processing problems for a 1-GHz band-idth broadband array with 1000 elements and 1000
apsyelement exceeds 1015 multiply accumulate op-erationsys just for the weight readout. The adapta-tion with LMS is at least as computationallyintensive, and covariance matrix estimation and in-version techniques typically used in digital adaptive-array processing are prohibitive for this case, inwhich the covariance matrix is 106 3 106 elements.
uch a matrix would require O~N3) 5 1018 operationsto initially invert, and then would require O(N2) 5012 operations to perturb the matrix inverse and
weights at each adaptation interval, desired to be asfast as microseconds, yielding 1018 operationsyshroughput requirements. To decrease this insur-ountable digital computational load, restrictive as-
umptions are often made on the number of signalsnd jammers, allowing for efficient subspace adapta-ion algorithms to be used, which perform acceptablyntil the number of signals becomes too large. An-ther approach is to employ subarray techniques thatield smaller numbers of digitized signals ~and cor-esponding adaptive degrees of freedom!, reducinghe digital computational load for simpler signal en-ironments. Thus restrictive assumptions on theumber of signals are often made to decrease theigital computation load ~allowing for more efficientubspace adaptivity!; or alternatively, subarray tech-
ooblipoa
nwsAp
niques with smaller numbers of adaptive degrees offreedom are used so that acceptable performance canbe obtained until the number of signals becomes toolarge. Unless the problem is simplified, these digi-tal computation loads exceed even the most optimis-tic performance projections for massively parallel,wafer-scale, and multi-chip-module digital signal-processing implementations, making this an appro-priate problem for an optical signal-processingsolution. In addition, rf photonic techniques allowfor wide instantaneous bandwidths of 1–2 GHz, lim-ited only by AO technology, which can be tuned overthe full 1–20-GHz rf band with only a single addi-tional electro-optic modulator ~EOM! with no mixersr downconverters. Fully photonic processing withnly one traveling-wave detector avoids noise contri-utions from the N detectors often required in paral-el photonic link systems, thereby potentiallymproving system dynamic range. Therefore the rfhotonic techniques presented here appear to be thenly viable solution for these next-generation fullydaptive, large, wideband array processing systems.
3. Optical System Analysis
In this section, we perform a detailed analysis of theBEAMTAP optical system. The analysis includesmany of the necessary complexities of double-sidebandamplitude modulation in EO devices, single-sidebandpolarization-switching diffraction in the AOD, finiteresponse time of the photorefractive crystals, read–write multiplexing and isolation with polarization andangle multiplexing, as well as spatial filtering of thediffracted orders, interferometric detection on the trav-eling fringes detector ~TFD!, and the time delays of thefeedback loop. This analysis not only validates theidealized algorithmic analysis presented in Subsection2.B but also provides valuable information on the ne-cessity of proper alignment of the interferometricbeams, the unavoidable bias buildup if a TDI CCD isused for the tap-in delay line, the difficulty with lowmodulation depths of the EO devices, and the proces-sor bandwidth limitation due to feedback delay30 andits mitigation with the compensating delay of thewriting beams. However, the analysis does neglectdiffraction, laser drift, and noise ~relative intensity,
Fig. 4. Optical architecture of BEAMTAP. A single coherent laser is divided with two beam splitters with amplitude reflectances ar anda9r, and amplitude transmittances at and a9t, respectively, to drive both the fiber-feed network and the BEAMTAP processor—the fiber-feed
etwork from the phased array is shown on the left-hand side. The diffracted light from the AOD interferes with signals from the array,hich are imaged through lens system L0 to form gratings in the photorefractive crystal ~PR crystal!. Diffraction of the phased-array
ignals off this grating is detected by a synchronous TFD, which has a carrier velocity matched to the magnified acoustic velocity of theOD by the lens systems L1 and L2—producing a resonant charge carrier distribution q~x2, t!. The output signal o~t! is amplified by g1,assed through a bandpass filter ~BPF!, subtracted from the desired signal d~t!, amplified by g2, and fed back into the AOD as the feedback
signal f ~t! to close the adaptive feedback loop necessary for the system to cancel any jamming signals present in the signal environment.A Rochon prism, linear polarizer ~LP!, and spatial filter ~SF! are used for the read–write isolation of the AOD beam from the diffractedphased-array signals off the grating. The illustrated system places the photorefractive crystal in the image plane of the fiber feed andthe AOD, and uses orthogonally propagating fields, for illustrative purposes.
10 January 2000 y Vol. 39, No. 2 y APPLIED OPTICS 217
faut
ffit
aaa
o
2
shot, thermal, etc.! for the sake of simplicity. Nu-merical simulations of the operation of the processorare then presented that demonstrate several of thecompensation mechanisms that are intrinsic to theadaptive holographic weight matrix.
A. System Overview
Figure 4 shows one possible optical architecture forthe implementation of the BEAMTAP system withthe photorefractive crystal in the image plane of theAOD. ~Alternative systems could use the photore-ractive crystal in a Fourier or Fresnel plane of eitherrm; the undiffracted beam from the AOD could besed as a reference; or an alternative read–write mul-iplexing scheme could be used29—the key require-
ment is that the AOD be imaged and velocitymatched onto the TFD.! In the illustrated system asingle powerful low-noise laser is divided by a beamsplitter ~amplitude reflectance ar! and frequencyshifted by an EOM to within 1 GHz of the rf centerfrequency ~typically in the range of 1–20 GHz!. Thisrequency-shifted beam is then distributed through aber-feed network and is modulated by EOM’s withhe signals sn~t! coming from each of the elements of
the phased-array antenna. The first frequency-shifting EOM in combination with the antennaEOM’s in the phased-array, operating at the rf centerfrequency, produce a frequency shift commensuratewith that produced by the AOD, allowing for process-ing of rf signals anywhere within the full 1–20-GHz rfband. The rf frequency domain for double-sidebandmodulation is illustrated in Fig. 5, although only onesideband is used in the analysis. Each of the rf-modulated signals is then transmitted from the mod-
18 APPLIED OPTICS y Vol. 39, No. 2 y 10 January 2000
ulators in the phased array through the fiber-feednetwork to an array of terminated fibers. These fi-bers can be in any topological arrangement ~1-D lin-ear array, 2-D hexagonal array, random, etc.! andwith an arbitrary permutation with respect to thespatial topology of the phased array, but for simplic-ity, in this analysis we assume that there is a 1-to-1mapping between an equispaced linear phased-arrayantenna and an equispaced array of optical fibers.The modulated signals from the fiber-feed networkare then imaged into the photorefractive crystal byuse of the 4f system L0.
The transmitted beam from the first beam splitter~amplitude transmittance at 5 =1 2 ar
2! is split inton interferometric reference beam ~amplitudeta9rEo! and an input to the AO Bragg cell ~amplitudeta9tEo!. The beam is expanded, collimated, and tilted
to the Bragg angle, whereby a feedback signal f ~t!applied to the AOD launches a propagating acousticwave that modulates the incoming light by means ofthe AO effect and diffracts it at the Bragg angle.Tangential matching in an anisotropic polarizationswitching device is used here so that the diffractedbeam has an orthogonal polarization to that of theundiffracted beam. The diffracted beam is then im-aged into the photorefractive crystal by use of the 4flens system L1, whereas the undiffracted beam isblocked in the Fourier plane of the lens system L1 byuse of a spatial filter. The diffracted beam from theAOD interferes with the spatiotemporally modulatedoptical field launched by the fiber-feed network, andthe time integration of this interference pattern pro-duces 45° gratings at positions corresponding to de-lays where the beam containing the desired signal
Fig. 5. Frequency offset scheme for the phased-array radar processor that allows for tuning of the processing bandwidth B ~determined bythe AOD bandwidth and typically limited to 1–2 GHz! anywhere within the rf spectrum spanned by the EOM’s bandwidths. The laser beamvl is premodulated by 6vs ~in this case only the negative sideband is used!, and then the rf signals modulate within a bandwidth of B andare offset by 6vp such that the desired mix term overlaps the bandwidth B and carrier vr of the AOD. The photorefractive crystal respondsnly to the near-dc interferometric products between the phased-array signals and AO diffractions, and all moving gratings wash out.
dab
oq
rtra
d~t! is well correlated with one of the beams from thefibers containing the signals from the antenna arraysn~t!.
Light from the fiber-feed network then diffracts offthe gratings in the crystal. The time delays ~both
ue to the inherent delays produced by the phased-rray antenna and to random time delays introducedy the small length mismatches between the fibers!
are transformed into diffracted beams at spatial po-sitions proportional to their respective time delays.These position-dependent time delays will be delayedand summed with appropriate time alignment by de-tection with a time-integrating traveling-wave detec-tor array matched to the velocity of the image of theAOD. Read–write multiplexing with angle and po-larization encoding is implemented by use of aquarter-wave plate, Rochon prism, polarizer, andspatial filter to isolate the diffracted readout beam ofthe phased array from the diffracted write beam ofthe AOD. Details of read–write multiplexing archi-tectures have been covered in previous publica-tions,17,31 so are delineated only briefly in this paper.
The diffracted signals are then imaged onto a time-integrating traveling-wave detector implementedwith a TFD21,22 where they interfere with the refer-ence beam from the laser—which must be at the ex-act angle that the undiffracted dc beam from the AODwould have occupied if it had not been blocked. Ateach resolvable temporal increment, the TFD is readout after the corresponding charge packet has tra-versed the length of the detector, accumulating anddelaying detected signal contributions from each lo-cation. The resulting output waveform is an adap-
tively weighted and coherently reconstructed versionof the signal.18 For closed-loop adaptive processing,this output signal o~t! is then subtracted from ourdesired signal d~t!, which is well correlated with thesignal of interest s~t! ~for instance, a known chirp canbe broadcast along with an unknown pseudorandomnoise waveform!. This difference signal generatesthe feedback signal f ~t! 5 d~t! 2 o~t!, which is then fedback into the AOD. The feedback loop provides thenecessary error-driven adaptation required for beamsteering the main beam toward the desired signalsource and nulling out any undesired jammerspresent in the signal environment at the input of thephased-array antenna—which can include both near-field and far-field, as well as narrow-band and broad-band, jammers.
B. Acousto-Optic Device
The single-sideband feedback signal amplified andapplied to the AOD transducer consists of the differ-ence between the processor output signal o~t! and thesteering rf signal d~t! such that f ~t! 5 g2@d~t 2 td! 2g1o~t 2 tf !# ~where g1 and g2 are the gains of thephotodetector output and the AOD power amplifier,respectively; td is a constant reference delay that ac-counts for the arbitrary timing of the desired signal toensure that the correlation peak with the arrivingarray signals is within the AOD aperture; and tf is thefeedback delay due to propagation through wires, am-plifiers, and the bandpass filter!. The signal isctave-bandwidth, limited around a rf center fre-uency vr, which is implicitly included in the analytic
signal representation f0~t!, f ~t! 5 f0~t!exp~2ivrt!,
Fig. 6. Example of a linear equispaced phased array and a random fiber-optic feed network. A broadband rf signal is incident from oneangle and a jammer j from another angle onto a phased-array antenna with element spacing d, creating time delays tp and tj between theelements of the array for the signal and jammer, respectively. Both signals are modulated onto a much-higher-frequency optical carrierand transduced into the optical domain; then these signals are propagated through an optical fiber-feed manifold to a linear array ofpolished fiber ends. While propagating through the polarization-maintaining fiber, each beam passes through a polarizing beam splitter~PBS! so that the py-polarized writing beam passes through the delay loop and experiences a time delay T with respect to the pz-polarizedeading beam. Typical fiber cores of 8 mm and spacings of 250 mm are shown being collimated by a lenslet array, which allows us to makehe simplifying assumption of propagation without diffraction used in the analysis of this section. These lenslets are not required in theeal system, and the simplifying assumption of propagation without diffraction is not required in practice, but is illustrated here fornalytic and diagrammatic simplicity.
10 January 2000 y Vol. 39, No. 2 y APPLIED OPTICS 219
p
B
o
w
AE
2
where fo~t! is a real bandlimited baseband signal.The signal will propagate through the AOD at theacoustic velocity vA in the positive x direction at a
lane x0 and time t0, resulting in a modulated output
f ~x0, t0! 5 g2FdSt0 2 td 2x0
vA2 tr1D
2 g1oSt0 2 tf 2x0
vA2 tr1DG
; fSt0 2x0
vA2 tr1D , (9)
where tr1allows for the AOD aperture to be centered
in the optical system.A collimated optical beam with an angular carrier
frequency vl, which is incident on the AOD at theragg angle uB 5 sin21 ukBuyuk0u, where uk0u 5 vlyc 5
2pyl0 is the free-space wave number and ukBu 5 vAycis the acoustic wave number, will diffract off theacoustic wave to produce a diffracted beam and anundiffracted beam
EA~x0, t0! 5 at a9t Eo exp@i~kB x0 2 vl t0!#
3 F~1 2 hAO2 u f u2!1y2px
1 hAOfSt0 2x0
vA2 tr1DpyG , (10)
where at 5 =1 2 ar2 is the amplitude transmittance
f the light from the beam splitter, hAO is the ampli-tude diffraction efficiency of the AOD per volt, andu f~t0 2 x0yvA 2 tr1
!u2 has been written as u f u2 in theundiffracted beam. Anisotropic diffraction in theAOD results in diffracted and undiffracted beamswith orthogonal polarizations, py and px, respectively.The undiffracted beam can be used as an interfero-metric reference that is automatically aligned at thecorrect angle to produce a nondispersive traveling-fringe pattern on the tap-out detector. The draw-back to this design is that this strong unmodulatedbeam will cause erasure and decrease modulationdepth in the photorefractive crystal, resulting in re-duced diffraction efficiency from the weight matrix.In addition, the undiffracted beam from the AODmay have too much residual modulation because ofits nonlinear dependence on the signal’s strength be-ing injected into the tap-in device. A clean referencebeam can be split off before the AOD and routed toavoid passing through the crystal before being rein-jected onto the TFD at the correct angle so that itinterferes with the diffracted signal beam to producea nondispersive traveling-fringe pattern, as shown inFig. 4.
The diffracted and the undiffracted beams are Fou-rier transformed by the first lens so that the undif-fracted beam can be blocked. The diffracted beam isthen inverse transformed by the second lens to pro-duce an image of the AOD in the crystal. Notingthat m1 5 x1yx0 is the magnification of lens systemL1, the electric field distribution can be approximated
20 APPLIED OPTICS y Vol. 39, No. 2 y 10 January 2000
within the photorefractive crystal. For the sake ofsimplicity, the effects that are due to diffraction of thepropagating fields are neglected in this analysis. In-stead, we account for propagation with a phase accu-mulation so that the field within the photorefractivecrystal is
EA~x, z, t0! 5 at a9t E0 exp@i~kB x 2 vl t0!#
3 hAO fSt0 2xv
2 tr1Dexp~ikz! py, (11)
where k 5 2pnyl0 is the wave-vector length in thephotorefractive crystal of index n and the velocity vahas been scaled by the magnification to a velocity v.A detailed analysis demonstrating that the effectsdue to diffraction are holographically compensatedfor by the processor has already been presented.32
C. Phased Array
Next, consider the fiber-optic modulation topologyshown in Fig. 6. A desired signal produced by abroadband far-field plane wave arriving on a linearequally spaced array at some angle us is illustrated.The signal at a particular element n will have a dif-ferential time delay between adjacent phased-arrayelements tp 5 dyc sin us, where d is the spacing be-tween the elements and c is the speed of light. Thedesired signal at the nth element of an equispacedarray can thus be described as sn
d 5 r~t 2 ntp!. Anyfar-field jammers present within the signal environ-ment will also experience incremental time delaysgiven by tj 5 dyc sin uj. If more than one jammer ispresent, the jamming signal at the nth element can bedescribed as s n
j 5 ¥j jj~t 2 ntj!. We will assume thatthe desired signal and any jamming signals are in afrequency band no greater than the bandwidth of theAOD centered around vp, but that this center fre-quency can be anywhere within the modulation band-width of the EOM’s. A single sideband of the signaldetected by the nth array element is thus
sn~t! 5 r~t 2 ntp! 1 (j
jj~t 2 ntj!, (12)
here the carrier term vp is implicit within thesingle-sideband analytic representations r~t! and j~t!.More generally, the ntp andyor the ntj term can in-stead be an arbitrary function of n to represent anyrandom time delays caused by positional errors in theplacement of the array elements, conformal arrays,near-field signals or jammers, or an arbitrary spatialpermutation of the topology of the array of fibers withrespect to the array elements.
Amplitude modulation by the first frequency shift-ing EOM before the phased-array offsets the lightcarrier vl by 6vs giving a new carrier vc 5 vl 6 vs~where we will choose to use only one of the sidebandsand reject the other with a fiber bandpass filter!.
fter it is amplitude modulated by the array of NOM antenna transducers in the fiber-feed network,
laGn
cafsfimlp
dbw
pa
rdao
t
the electric field distribution launched into the pro-cessor by the fibers is
EP~z0, t0! 5ar
ÎNE0 (
n51
N
exp$2i@vc~t0 2 tn!#%a~z0 2 nD0!
3 @Î1 2 hP2 u snu2 1 hPsn~t0 2 tn 2 T!
1 h*Ps*n~t0 2 tn 2 T!# py, (13)
where usnu2 5 ur~t0 2 ntp 2 tn 2 T! 1 (j jj~t0 2 ntj 2tn2T!u2. In Eq. ~13!, ary=N is the amplitude of theight after equal splitting into each of the N fibers.~z0 2 nD0! is the aperture function ~typically aaussian distribution with a width of 5–9 mm! of theth fiber, located at position nD0 where D0 is the
spacing between the fibers. For fibers in a siliconV-groove array, a typical value of D0 might be 250mm. hP is the amplitude-modulation efficiency ofthe EOM’s per volt and is assumed to be the same forall modulators. tn is a random time delay that ac-counts for phase and time shifts due to fiber lengthmismatches, which can change with temperaturevariations and shifting antenna configurations.These mismatches affect both the optical carrier vc~for submicrometer mismatches! and the microwavearrier vr ~for centimeter mismatches! and must beccounted for in the formalism. T is the round-tripeedback loop time delay, which, as described in Sub-ection 2.B, is necessary to ensure causality of thenal output signal o~t!. This time delay is imple-ented on each phased-array signal by use of a po-
arizing beam splitter in each fiber, which causes theˆ y-polarized writing beam to pass through a fiber-delay loop while the pz polarized reading beam prop-agates directly through the fiber-feed network.Polarization-preserving fiber is used to to ensure thatthe polarizations of these two beams are maintainedwhile propagating though the fiber so that the outputwriting beam will have the same polarization as thediffracted beam from the AOD py. Thus Eq. ~13!
escribes the electric field distribution of the writingeam while the reading beam will be pz polarizedithout the delay T.It is vital to realize that these lengths can easily be
erturbed by small motions of the fibers or by temper-ture changes ~thereby giving rise to tn!, and although
such fluctuations will have no effect on the rf carrierphase, the optical phase can swing wildly and must becompensated through adaptation. In the system pre-sented in this paper, adaptation will occur in a timewell below 1 ms ~much faster than the photorefractivehologram response time17!, which should be sufficientto compensate for these phase fluctuations. We arerepresenting the EOM as being operated as an ampli-tude modulator, but for the sake of simplicity the com-plex conjugate term will be subsequently dropped toproduce a single-sideband phase modulator. Ideally,we would use single-sideband suppressed carrier mod-ulators, which would eliminate the dc term as well.Including these terms in the analysis yields essentiallythe same overall result, since they are at the wrongfrequency to record stationary gratings and thus will
not affect the photorefractive crystal except throughincoherent erasure. However, in addition to the pre-viously noted considerations, the phase in the py-polarized fiber-delay loops must be stabilized withrespect to the undelayed pz-polarized phases for agiven fiber. Although this can be accomplished withindividual polarization-locking feedback-to-loop fiberstretchers, a single novelty filter33 can be used insteadto stabilize all of the fiber loops simultaneously ~and toemove the unwanted dc term as well!. We leave theetails of this loop stabilization to a subsequent papernd henceforth assume a phase-stable delay by T in allf the fiber loops.This phased-array signal beam is then imaged
hrough lens system L0, and noting that m0 5 z1yz0 5D1yD0, the electric field distribution within the pho-torefractive crystal can be described when we includea propagation term that represents phase accumula-tion ~in the negative x direction!
EP~x, z, t0! 5ar
ÎNE0 (
n51
N
exp$2i@vl~t0 2 tn!#%a~z 2 nD!
3 @~1 2 hP2usnu2!1y2 1 hP sn~t0 2 tn 2 T!#
3 exp~2ikx! py, (14)
where the offset frequency of vs has now been incor-porated implicitly into sn~t0!. We will choose themodulation frequencies such that vr 5 vs 1 vp, so theresulting modulation produced by the EOM’s over-laps with the Doppler frequencies produced by theAOD, as shown previously in Fig. 5.
D. Grating Formation
As the optical beam from the AOD interacts with thebeam from the phased array, and if the frequency spec-tra of the two beams overlap, an index grating isformed within the photorefractive crystal, owing to thestationary interference pattern. With first-orderanalysis, the grating evolution can be approximated by
G~x, z, t9! 5 b *2`
t9
EA~x, z, t0!E*P~x, z, t0!
3 expF2~t9 2 t0!
t Gdt0 1 c.c., (15)
where b is the sensitivity of the crystal ~cm2yJ!, t isthe photorefractive time constant, and the details ofthe photorefractive dynamics have been idealizedfrom their much more complex form.34 For times t9.. t the convolution with exp~2t0yt! results in acausal Lorentzian low-pass filter in the temporal fre-quency domain with a width Df 5 1y~2pt!, typically inthe range of hertz to kilohertz. Thus only stationarygratings can be written in the photorefractive crystalbetween equally Doppler-frequency-shifted beams,which are then weighted by the photorefractive timeconstant t, which in turn is inversely proportional tothe dc intensity level t } toyIdc, where Idc 5 uEA~x, z,t!u2 1 uEP~x, z, t!u2. The spatial-frequency responseof the photorefractive crystal selects only high-
10 January 2000 y Vol. 39, No. 2 y APPLIED OPTICS 221
ts
w
and
v
2
spatial-frequency gratings to be recorded, althoughdc terms do lead to grating erasure.35 Thus the pho-torefractive grating is given by
G~x, z, t9! 5 b *2`
t9 HFexp@ik~z 1 x!#exp~ikB x!
3ar at at9
ÎNE0
2hAOfSt0 2xv
2 tr1DG3 F(
n51
N
exp~2ivl tn!a*~z 2 nD!
3 @~1 2 hP2usnu2!1y2 1 hPs*n~t0 2 tn 2 T!#pyGJ
3 expF2 ~t9 2 t0!
t Gdt0. (16)
Since the unmodulated EOM beam produces a mov-ing grating with the diffracted AOD beam at the rfcarrier frequency vr, this grating washes out, owingo the low-pass nature of the photorefractive crystal,implifying the expression of the grating to
G~x, z, t9! 5 exp@ik~z 1 x!#exp~ikB x!
3 k0(n51
N
exp~2ivl tn!a*~z 2 nD!
3 *2`
t9
fSt0 2xv
2 tr1Ds*n~t0 2 tn 2 T!dt0,
(17)
here k0 5 bEo2tarata9ty=N hAOhP.
The phased-array signal environment consists of therf signal of interest r~t! and various jammers jj~t!.Therefore the desired signal d~t! is chosen ~or gener-ated adaptively27! to have good correlation properties
with r~t! and poor correlation with the jammers so thatcross correlation is formed between the feedback sig-al f ~t0 2 xyv 2 tr1
!# and s*n~t0 2 tn 2 T!, the signalsfrom each element of the phased array. The crosscorrelation results in 45° localized gratings, corre-sponding to each resolvable time delay between adja-
22 APPLIED OPTICS y Vol. 39, No. 2 y 10 January 2000
cent antenna elements, as indicated by the exp@ik~z 1x!# term wherever d~t0 2 td 2 xyv 2 tr1
! overlaps withs*n~t0 2 tn 2 T! within the crystal. The gratings,whose amplitudes correspond to a weight magnitude,are written with a complex phase factor ~correspond-ing to the phase of the complex weights and physicallymanifested as a shift of the high-frequency grating!that accounts for optical propagation delays and rfsignal delays at positions corresponding to those re-spective time delays. Assuming the time delay dis-tortions of the fiber feed network are small ~theariations of tn over n are less than the inverse band-
width 1yB!, the envelope of the 45° localized gratingswill lie on a tilted line, whose tilt angle corresponds toa specific angle of arrival ~AOA!. Overall time delayscan be accounted for by shifts of this line, and changesin the AOA are manifested as changes of the tilt angleof the envelope of the gratings. The gratings withinthe cross-correlation envelope evolve as a function oftime, allowing the processor to adapt to changing rfsignal environments as well as slow ~kilohertz or slow-er! phase changes that are due to temperature or me-chanical fluctuations within the fiber-feed network.Small phase drifts simply shift the phase of the grat-ings within the envelope, whereas larger changes ofthe random fiber delays will shift the position of theenvelope of the gratings.
E. Diffraction of the Phased-Array Signal Beam off theGrating
As light from each of the phased-array signal fiberspropagates through the crystal, picking up a phasefactor exp~2ikx!, each beam of light interacts withthe local gratings and diffracts off of it. An addi-tional phase of exp@ik~L 2 z!#, where L is the lengthof the crystal, is accumulated as each of the diffractedbeams propagate along z to the face of the crystal.As a result, the diffracted beam at the edge of thecrystal can be calculated by
where we have use the expression for the undelayedversion of the phased-array field amplitude to diffractoff the grating. Note that the phase accumulated bypropagating in the negative x direction exp~2ikx! iscanceled with the x dependence of the phase within thegrating exp@ik~z 1 x!#; and at each z the phase accu-mulated by propagation to the output exp$i@k~L 2 z!#%
after diffraction off the grating cancels with the phaseof the grating dependent on z. As a result, each of thecomponents of light that diffracts off the grating fromthe fibers picks up the same overall phase exp~ikL! andtherefore adds coherently. Because the fiber profilesdo not overlap, the undiffracting fields are spatiallyorthogonal to one another so that n 5 n9 and the twoums collapse, which allows the random phase delaysresent in the carrier exp~ivltn9! to be canceled withhe holographically encoded phase exp~2ivltn!.
The signal strength received from the phased-arrayantennas will usually be small, and electrical pream-plification may be required at each element of the an-tenna before optical modulation, but even with
amplification the modulation depth of the EOM is typ-ically ,,1%. Thus there are two consequences of thec term: erasure of the desired grating—thereby de-reasing the grating strength ~and therefore the nullepth!, and unwanted diffraction of the dc term fromach fiber off of the correlation grating from thatber—thereby producing unwanted bias when de-ected on the TFD. The dc term in Ep that diffracts off
the grating ~although with an incorrect phase frontthat is due to the fiber time delays! could potentially bemuch stronger then the modulated term, creating sucha strong bias that it could swamp out the signal ofinterest. The actual diffraction efficiency of the un-modulated dc terms from the phased array off the grat-
Fig. 7. k-space representation of the polarization, angle, and time-multiplexed recording and readout geometry in strontium bariumniobate ~SBN! used to separate the diffracted phased-array beam kd from the AOD beam kA used to write the grating. The uppereft-hand portion of the figure is a 2-D projection of the three-dimensional momentum space on the right-hand side, where the three beamsncident on the photorefractive ~the write beam from the phased array kP1
with polarization py, the vertically deflected read beam fromthe phased array kP2
pz, and the diffracted AOD beam kApy! refract into the crystal and are projected onto the ordinary and extraordinaryomentum surfaces, based on their respective polarizations. The deflected read beam has been tilted vertically in the direction of Bragg
egeneracy, allowing for efficient light diffraction off the grating kG produced by the interference between kP1and kA. This produces a
diffracted beam kd with polarization px ~once refracted back into air!, which is vertically tilted from the reference signal beam from theAOD kA with polarization py, as can be seen in the bottom portion of the figure. A lens, when in conjunction with a spatial filter, can thenbe used to block the beam from the AOD while allowing for the diffracted beam to pass through the system. A polarizer can also be usedto increase the overall isolation between the two beams.
10 January 2000 y Vol. 39, No. 2 y APPLIED OPTICS 223
bdwus
af
mwp
par
te
b
at
2
ings will depend on the details of the time delays tn, butecause of coherent superposition of the diffractions offifferent parts of the gratings, the diffraction efficiencyill tend to be zero for large array sizes. To avoid anynwanted diffraction of the strong dc terms, the EOM’should use a suppressed carrier modulation scheme36
~or a photorefractive novelty filter could be used at thephased-array fiber-feed outputs to remove the dc car-rier term, as mentioned previously33!. We will thusassume that the dc term is not present in Ep. Finally,noting that *0
Lua~z 2 nD!u2dz 5 1 is just the total nor-malized optical power present in the nth fiber,
Ed~x1, t9! 5 exp@i~kB x1 2 vl t9!#exp~ikL!
3 Eok1(n51
N
sn~t9 2 tn! *2`
t9
fSt0 2x1
v2 tr1D
3 s*n~t0 2 tn 2 T!dt0pz, (19)
where k1 5 k0arhPy=N. Equation ~19! shows thatthe diffracted output from the photorefractive crystalconvolves the cross correlation between the feedbacksignal and the detected signal with the signals fromthe phased-array antenna, with the detected signalused in the cross correlation being delayed by theround-trip delay time T to ensure causality. Notethat the random phase factors that were originallypresent in the beam have been holographically com-pensated for and have been removed from the dif-fracted signals. Over a region defined by thetransverse width of the grating envelope we will see atraveling-wave version of the signals from the variouselements of the phased array propagating at a velocityv independent of the arrival angle of the signal.
F. Read–Write Isolation
As described so far, the readout diffracted beam fromthe phased array is not separable in angle from themodulated write beam from the AOD. For our adap-tive phased-array processor, it is important to be ableto separate these two beams so that the writing beamcan be completely blocked from the detector plane.Otherwise, any leakage of the writing beam onto theTFD will limit the available loop gain and, therefore,the processor null depth.29 This can be accom-plished in many ways by use of polarization, timing,or angle multiplexing—although the isolation ratiosrequired ~well in excess of 60 dB! are not achievablewith polarization multiplexing by itself. Here weconsider a scheme that first vertically ~i.e., orthogonalto the interaction plane of Bragg selectivity! deflectsthe undelayed portion of the light from the phased-array fiber feed by a Rochon prism ~with a smalldeflection angle ur of the order of 1°!. As discussedbove, the x-propagating writing and reading beamsrom the array feed have orthogonal polarizations ~ py
and pz, respectively!, because of the polarizing beamsplitter used in the fiber-feed network to ensure thatthe writing beam propagates through the fiber loopand is delayed by T with respect to the reading beam.The polarizations of the undeflected beam py and the
24 APPLIED OPTICS y Vol. 39, No. 2 y 10 January 2000
deflected beam pz from the Rochon prism are shownin the k-space diagram in Fig. 7 and are designated askp1
and kp2for the writing and the reading beams,
respectively. The reading beam is tilted vertically inthe direction of Bragg degeneracy, allowing for effi-cient diffraction off the original grating kG, whichwas written between the two py-polarized beams kP1
and kA, the beams from the phased array and theAOD, respectively. This produces a diffracted beamkd, which is vertically tilted from the reference signalbeam from the AOD kA so that in the Fourier plane ofthe first lens in lens system L2 ~with focal length F2!they are spatially separated by yr 5 tan urF2, and thebeam kA that wrote the grating can be blocked witha spatial filter.17 A linear polarizer with the trans-
ission axis in the x direction is also used to block theriting beam from the AOD ~which is polarized in the
ˆ y direction! to increase the overall isolation ratiobetween the reading and the writing beams. Thisscheme requires that an xz-polarized beam ~withinthe crystal! be efficiently diffracted into an xz-
olarized beam at a different angle, which is actuallyconvenient geometry for high-diffraction-efficiency
eadout using r33 in crystallographically cut stron-tium barium niobate ~SBN! with the c axis at a 45°angle with respect to the x and the z axes, as shownin the figure. Alternatively, an xz-polarized beamcould be diffracted into a zx-polarized beam by use ofhe conventional geometry for high-diffraction-fficiency readout using r42 in barium titanate
~BaTiO3! ~either crystallographic or 45° cuts35,37!.At this point, the off-axis interferometric reference
eam ~at the laser frequency vl! needs to be rein-jected. Its position should correspond to the positionin the Fourier plane ~shifted vertically by yr!, wherethe dc beam from the AOD would have been imagedto if it had not been blocked in the first Fourier planein lens system L1. This can be done by placement ofa fiber carrying the reference beam with amplitudeatar9E0 at the correct position in the Fourier plane orby use of the appropriate pick off mirrors, as shown inFig. 4. A final lens can then be used to retransformthe diffracted beam coming from the phased array inthe photorefractive crystal, and the injected inter-ferometric reference beam, onto the TFD.
G. Interferometric Traveling-Fringes Detector
Assuming a magnification of the 4f system of m2 andrealignment of the system axis in the vertically
ilted plane, the total field at the TFD is
Ed~x2, t9! 5 E0 exp@i~kB x2 2 vl t9!#
3 Fat ar9 1 hDk1 (n51
N
sn~t9 2 tn!
3 *2`
t9
fSt0 2x2
vD2 tr1Ds*n~t0 2 tn 2 T!dt0Gpx,
(20)
tt
vt
D
Tl
t
iti
where m2 5 x2yx1, hD is the overall amplitude trans-mission of the diffracted beams through the read–write multiplexing architecture, vD is the velocity vscaled by the magnification m2 so that it matches thedetector velocity, and we have dropped the constantphase term exp~ikL! without any loss of generality.
The intensity at the detector is given by
I~x2, t9! 5 ~at a9r E0!2 1 ~hDk1!
2U(n51
N
sn~t9 2 tn!
3 *2`
t9
fSt0 2x2
vD2 tr1Ds*n~t0 2 tn 2 T!dt0U2
1 k2 (n51
N
sn~t9 2 tn! *2`
t9
fSt0 2x2
vD2 tr1D
3 s*n~t0 2 tn 2 T!dt0 1 c.c., (21)
where k2 5 atar9hDk1.The TFD is based on the synchronous drift of pho-
ogenerated carriers with a moving interference pat-ern21,22 generated by interfering two beams of light
at different frequencies—the diffracted beam fromthe phased array and the interferometric referencebeam, and the device is also used to time delay the rfsignals. At a given moment in time, the spatiallymodulated light that is incident on the photoconduc-tive layer of the detector will generate photocarriersin proportion to the local intensity, and these carrierswill drift with a velocity proportional to the appliedbias voltage. Varying the applied bias voltage in thephotoconductor changes this drift velocity, and a pho-tocurrent resonance peak occurs when the inducedelectron drift velocity equals the fringe velocity of themoving interference pattern. For the desiredphased-array signal sn~t9! diffracted by the grating,the interferometric fringes will all move at the mag-nified acoustic velocity, which can be set equal to thevelocity of the photogenerated carriers by fine tuningthe bias voltage. The unwanted but unavoidablediffractions of the jammers off the sidelobes of thephotorefractive grating, however, will not have thenecessary frequency–angle relationship to produceconstant velocity moving fringes at the carrier andthe drift velocity. Thus the jammers will be some-what suppressed and washed out—both by the angu-lar selectivity of the diffraction by the photorefractivecrystal, and in the frequency domain by the temporalfiltering associated with resonant detection of con-stant velocity interferometric fringes on the TFD.
The first two terms in Eq. ~21! are unwanted dcterms and will contribute only to bias in the detector.The desired detected interference pattern, however,is moving at a velocity vD 5 m1m2vA, the acousticelocity of the AOD scaled by the overall magnifica-ion of the lens systems L1 and L2. If the AOD has
a time–bandwidth product M, with spatial resolutionx2, then the TFD will detect the moving interference
pattern over a time TD 5 MDx2y~m1m2vA! 5MDx2yvD with a responsivity R. Assuming that themagnification has scaled the velocity and been fine
tuned with the bias voltage across the device to besynchronous with the photogenerated carriers, thefinal output is
o~t! 5 R *2TDy2
TDy2
*2`
t
I~x2, t9!dF~t9 2 t! 2x2
vD1 tr2G
3 dt9dx2yvD
5 R *2TDy2
TDy2
I~x2, t 1 tx 2 tr2!dtx, (22)
where tr2is another reference delay that allows the
FD to be centered within the optical system and theimits of integration to remain causal, and tx 5 x2yvD.
When we substitute this expression into the detectedsignal of interest, then
o~t! 5 Rk2 *2TDy2
TDy2
(n51
N
sn~t 2 tn 1 tx 2 tr2!
3 *2`
t1tx2tr2
f ~t0 2 tx 2 tr1!s*n~t0 2 tn 2 T!dt0dtx
5 Rk2 *2TDy2
TDy2
(n51
N
sn~t 2 tn 1 tx 2 tr2!
3 *2`
t
f @t1 2 ~tr11 tr2
!#s*n~t1 2 tn 2 T 1 tx 2 tr2!
3 dt1dtx. (23)
The feedback signal f @t1 2 ~tr11 tr2
!# contains theoutput signal with a feedback time delay tf, whichwhen combined with the reference delays ~tr1
1 tr2!,
provides the total round-trip time delay T 5 tf 1 ~tr11
r2!. The reference delays were used to center the
AOD and the TFD within the optical system, and~tr1
1 tr2! represents the total fixed time delay that is
ncurred by each of the signals propagating throughhe two TDL’s. Expanding the feedback signal intots desired and output signal components,
o~t! 5 Rk2 *2TDy2
TDy2
(n51
N
sn~t 2 tn 1 tx 2 tr2!
3 *2`
t
g2@d~t1 2 td! 2 g1o~t1 2 T!#
3 s*n~t1 2 tn 2 T 1 tx 2 tr2!dt1dtx
5 Rk2 *2TDy2
TDy2
(n51
N
sn~t 2 tn 1 tx 2 tr2!
3 *2`
t2T
g2$d@t2 1 ~T 2 td!# 2 g1o~t2!%
3 s*n~t2 2 tn 1 tx 2 tr2!dt1dtx. (24)
10 January 2000 y Vol. 39, No. 2 y APPLIED OPTICS 225
bat
r
w
u
f
3
tons
2
Equation ~24! shows that, with the total round-tripdelay T appropriately incorporated into both the feed-ack signal and the writing signal from the phasedrray, a temporal equation is created that is identicalo that of Eq. ~8!. A cross correlation is formed be-
tween fixed delayed versions of the signal from theantenna array and a delayed version of the feedbacksignal, which is then convolved with the instanta-neous signals from the array to provide the final out-put o~t!. Since o~t! appears on both sides of thisintegral equation, it is difficult to solve in the timedomain; hence we will subsequently transform to aFourier-domain representation.
H. Frequency-Domain Analysis
Equation ~24! represents the adaptation of the outputto an arbitrary signal environment with an arbitrarynumber of jamming signals present, where sn~t! 5˜~t 2 ntp! 1 ¥j jj~t 2 ntj!. The steady-state solutionof this equation can be derived by transformation tothe frequency domain and by algebraic manipulationof the spectra to solve for O~ f !. At steady state witht .. T, the output signal is a spatiotemporal convo-lution of the signal from the phased array with thecross-correlation function of the delayed feedback sig-nal with the delayed version of the phased-array sig-nal. When transformed to the temporal Fourierdomain, Eq. ~24! becomes
O~ f ! 5 Rk2 (n
Sn~ f !g2$D~ f !exp@i2pf ~T 2 td!#
2 g1O~ f !%S*n~2f ! p TD sinc~TD f !, (25)
where p denotes a convolution. Solving for thefrequency-domain output O~ f !,
O~ f ! 5
g2Rk2 (n
uSn~ f !u2D~ f !exp@i2pf ~T 2 td!# p TD sinc~TDf !
1 1 g1g2Rk2 (n
uSn~ f !u2 p TD sinc~TDf !,
(26)
here the exp@i2pf ~T 2 td!# phase term denotes thearbitrary timing necessary to ensure that the crosscorrelation between the desired signal and the sig-nals from the phased array occurs within the pho-torefractive crystal. Noting that the Fouriertransform of the input signal sn~t! is Sn~ f ! 5R~ f !exp~2i2pntp! 1 ¥j Jj~ f !exp~2i2pntj! and isdominated by the jammers, the term in the denomi-nator will simplify to the sum of the powers of thejamming signals present ¥nu¥j Jj~ f !u2 5 ¥n ¥juJj~ f !u2,since the jammers are assumed to be mutually inde-pendent. This is a valid approximation for manysignal environments, since typically the signal of in-terest is buried beneath the noise floor, whereas thejamming signal may be as large as 30–60 dB abovethe noise. The frequency-domain sinc function rep-resents the limitation of temporal degrees of freedom
26 APPLIED OPTICS y Vol. 39, No. 2 y 10 January 2000
but can be approximated as a delta function when TDis much greater than the correlation time of the de-sired signal,38 yielding
O~ f ! 5
gg1
(n
Sn~ f !$S*n~2f !D~ f !exp@i2pf ~T 2 td!#%
1 1 g (n
(j
uJj~ f !u2, (27)
where g 5 g1g2Rk2TD is the net gain around the loop.Spatial Fourier transformation can now be used to
illustrate the spatiotemporal frequency response of thesystem. Noting that Sn~ f ! 5 ¥l S~kl, f !exp~iklnd!,and that S~kl, f ! 5 ¥n Sn~ f !exp~2iklnd!, where kl 52ply~Nd! ~for l 5 2Ny2, . . . , Ny2!, and expanding thedetected signal into its constituent signal and jammercomponents at their respective angles of arrival,Sn~ f ! 5 R~ f !exp@i2pfyc sin~ur!nd# 1 ¥j Jj~ f !exp@i2pfyc sin~uj!nd# since tp 5 fycd sin ur and tj 5 fycd sin
j, yields
S~kl, f ! 5 FR~ f !dSkl 2fc
sin urD 1 (j
Jj~ f !
3 dSkl 2fc
sin ujDG p Nd sinc~Ndkl!. (28)
Thus broadband signals lie on a tilted locus in spa-tiotemporal Fourier space ~pivoting through kl 5 0,
5 0! and are blurred by the finite array size.The detected signal of interest R~ f !d~kl 2 fyc sin ur!
correlates against the desired signal D~ f !, and can beseparated into a known component aD~ f ! that corre-lates perfectly with D~ f !, and an unknown compo-nent a9D9~ f ! that is independent of it. This allowsfor further simplification of the output, since only thepart that is correlated with the desired signal willwrite a grating within the photorefractive crystal.Since the desired signal is also uncorrelated to any ofthe jamming signals, the output takes on the form
O~ f ! 5 (kl
S~kl, f !
gg1
auD~ f !u2 exp@i2pf ~T 2 td!#dSkl 2fc
sin urD p b~kl, f !
1 1 g (kl
(j
uJj~ f !u2dSkl 2fc
sin ujD p b~kl, f !
,
(29)
where b~kl, f ! 5 Td sinc~Tdf !Nd sinc~Ndkl! denoteshe total blur function. The linear adaptive-arrayutput-frequency response is given by a weighted sig-al spatiotemporal frequency spectrum that isummed across all spatial frequencies,
O~ f ! 5 (kl
S~kl, f !T~kl, f !, (30)
Ttttc
bp
and this allows the steady-state spatiotemporaltransfer function of the BEAMTAP processor to bedefined as
ics
The maximum array sensitivity is along the tiltedlocus in spatiotemporal frequency space correspond-ing to the desired signal AOA, with a bandwidthtuned to the power spectrum of the steering signaluD~ f !u2 ~actually flattened by the LMS dynamics!.
his output achieved the full array gain of Nd, sincehe beam-forming operation has a linear phase thatime delays the arriving signal to time align it withhe known reference signal. Jammer nulling is ac-omplished by an inverse filter ~the denominator!
that acts as a power nuller to any jamming signalspresent in the signal environment at their respectiveangles of arrival @denoted by the d~kl 2 fyc sin uj!term# and frequency spectrums. The total nulldepth of the processor is limited by the effective loopgain g, and the available null depth is distributedamong the number of jamming signals present.29
The grating within the photorefractive processor will
respond more quickly to higher-power jamming sig-nals to create a greater null depth, whereas smaller-power jamming signals will not be nulled as deeply,or as quickly ~although the jammers’ final power wille the same after adaptation!. As in all LMS-basedower nulling processors, when the signal of interest
s strong, its corresponding output amplitude will belamped to the signal power of the desired signal,ince the ¥kl
uS~kl, f !u2 term in the denominator of thetransfer function will no longer be dominated by thesum of the power of the jamming signals.
Beam steering is driven by the known componentad~t! of the desired signal detected at the array r~t! 5ad~t! 1 a9d9~t!, which can be either time multiplexedor code multiplexed with the spectrally overlappingunknown desired signal a9d9~t!. The TFD outputcontains both d~t! and d9~t! weighted proportionallyto a and a9, respectively, since they are broadcast bya single transmitter and arrive on the array withidentical phase fronts. At steady state, the ampli-tude of the known component at the differencing nodeadaptively adjusts to a nearly exact match of theknown reference signal; thus just after the differenc-ing node, the known component is canceled, and the
Fig. 8. Spatiotemporal Fourier space representation of the inputsignals used in the computer simulation of the BEAMTAP algo-rithm, which were selected to test the system’s jammer-nullingcapacity over a diversified signal environment. The desired sig-nal is a broadband Gaussian chirp ~1.2–1.8 GHz! at 0.25 rad AOA.Jammer 1 is broadband filtered Gaussian noise ~0.5–2.5 GHz! at20.2 rad AOA, and jammer 2 is a single-frequency 0.8 GHz at 0.5rad AOA sine wave, each of which are 30 dB stronger than thesignal of interest.
Fig. 9. Diagram illustrating the simulation of the BEAMTAParchitecture. The input signal is represented by the array of timehistories shown on the left-hand side. At every time step aninstantaneous slice of the input is detected by the antenna arraysand is propagated through the adaptive weight matrix ~center offigure!. The product of the input vector with the weight matrix isdiffracted vertically and detected and accumulated on the scrollingdetector ~top of figure!. The output o~t! is subtracted from thedesired signal d~t!, generating the feedback signal f ~t!, which is fedthrough the scrolling delay line ~bottom of figure!. An outer prod-uct between the scrolling feedback signal and a delayed version ofthe input is used to adapt the weights, producing the resultingtilted cross-correlation slice seen in the weight matrix in the centerof the figure at steady state.
T~kl, f ! 5
gg1
auD~ f !u2 exp@i2pf ~T 2 td!#dSkl 2fc
sin urD p b~kl, f !
1 1 g (kl
(j
uJj~ f !u2dSkl 2fc
sin ujD p b~kl, f !
. (31)
10 January 2000 y Vol. 39, No. 2 y APPLIED OPTICS 227
q
2
unknown desired signal a9D9~ f ! remains detectedwith the full array gain and all of the jammers areoptimally extinguished.
I. BEAMTAP System Simulations
We simulated the BEAMTAP algorithm to demon-strate its operation for broadband squint-free beamforming and jammer nulling and to verify the spatio-temporal frequency-domain transfer function formal-ism. Figure 8 characterizes the input signals inspatiotemporal Fourier space with transverse spatialfrequency along the vertical axis and a single-sidedtemporal frequency along the horizontal axis. Thedesired signal is a broadband Gaussian apodizedchirp whose 1ye spectrum spans the frequency rangefrom 1.1 to 1.9 GHz. Its AOA is set at 0.25 rad, andit falls on a tilted locus in spatiotemporal frequencyspace. Jammer 1 is a broadband filtered Gaussianwhite-noise signal, spanning the frequency rangefrom 0.5 to 2.5 GHz, with an AOA of 20.2 rad. Jam-mer 2 is a narrow-band sine wave, set at a frequencyof 0.8 GHz and with an AOA of 0.5 rad. Jammer 2 isselected such that it lies on the maximum of the firstsidelobe of the receptivity pattern of the chirp whenbeam forming is performed without jammers. In thesimulations presented here, the power in each jam-mer is 1000 times stronger than the power in thedesired signal. However, the power of all signalswas normalized in Fig. 8 for illustrative purposes tomake all the signals visible.
The diagram shown in Fig. 9 illustrates the variousfacets of the simulation. The leftmost figure shows
Fig. 10. AOA versus frequency receptivity pattern that developsafter adaptation when only the desired signal is present at theinput. Note that the main lobe at 0.25 rad AOA does not vary itsposition with frequency ~although its width does change! as itspans the entire input signal bandwidth—thereby demonstratingsquint-free TTD beam forming.
28 APPLIED OPTICS y Vol. 39, No. 2 y 10 January 2000
the spatiotemporal rf field amplitude detected by thearray antennas sampled at 125 ps ~i.e., with a fre-uency fs 5 8 GHz! along the horizontal time axis and
spatially sampled by the 64 antenna elements alongthe vertical axis. Note that only the jammers arevisible, since they are much stronger than the signal.The final weight values after adaptation for 300 mswith a net loop gain of 1 are shown in the center ofthis figure. Note also that, even though the signal istotally buried by the jammers, the dominant featurethat builds up in the weight matrix is the tilted stripethat results from the correlation between the desiredcomponent of the feedback signal f ~t! 5 d~t! 2 o~t!~the difference between the desired chirp signal andthe output! and the input signals sn~t! ~chirp andjammers! whose tilt is an indication of the AOA of thedesired signal. Weak sidelobes seen in the weightmatrix are due to a cross correlation between thesignal and the jammers and are responsible for themanipulation of the antenna nulls to point towardany undesired jammers. The instantaneous inputsignal vector from the array is multiplied ~e.g., dif-fracted vertically! by this weight matrix to transformarray element positions into topologically orderedspatial positions arranged with linear time delays atthe velocity of the TFD; detection on the TFD therebycompensates for the unknown time delays to each
Fig. 11. AOA versus frequency receptivity pattern after adapta-tion when the desired signal and strong jammers are both presentat the input—demonstrating squint-free jammer suppression withdeep nulls. Note the extremely narrow constant angle squint-freenull at the angle of the broadband jammer ~20.2 rad! over its fullbandwidth, a narrow-band null at 0.5 rad and 800 MHz with deepsidelobe nulls, and a slight reduction in the bandwidth of thesystem response to the desired signal in comparison with Fig. 10~although more than the full 1ye bandwidth of the signal is stilluniformly detected!.
actgaawl
assptrcssgGa
w
acjt
bfcataiBsWtutnaStn
ds
antenna element of the desired signal. The dif-fracted signal is deflected vertically where it is de-tected and accumulated in the scrolling detectorarray, which in this simulation was shifting at a sam-pled rate of 8 GHz. The upper part of the figureshows the contents of the TFD at this instant, illus-trating the accumulation of the light diffracted by thephotorefractive weights, which generates the systemoutput o~t! after propagating fully across the detectorperture. The lower part of the figure shows theontents of the scrolling input modulator ~AOD! athis instant in time. An outer product ~e.g., holo-raphic interference! between this signal in the AODnd a delayed version of the input signal is used todapt the weights. This simulation used an arrayith 64 antenna elements and 64 taps in the delay
ine.Figures 10 and 11 show the receptivity pattern asfunction of AOA and temporal frequency of the
ystem for two different cases, first with only theignal present in the environment, and then in theresence of the strong jammers as well. The recep-ivity patterns were produced by simulation of nar-ow Gaussian pulses incident on the phased arrayoming from different AOA’s, according to the expres-ion T~u, f ! 5 O~ f !yG~ f !ugn~u,t!, where G~ f ! repre-ents the spectrum of the incident Gaussian pulse,n~u, t! 5 exp@2~t 2 ndyc sin u!2yD2# represents theaussian pulse detected by each antenna element,nd D represents the 1ye pulse width. Each pulse is
propagated through the BEAMTAP system ~with fro-zen weights! to provide the corresponding output.The output is Fourier transformed and is then di-vided by the spectrum of the initial Gaussian pulse,providing us with the transfer function for a givenAOA. Repeating this process over several AOA’sgives the final transfer function as a function of u andf in terms of the weights
T~u, f ! 5 (n51
N
(t50
M21
Wtn expF2i2pfStst 1 ndc
sin uDG ,
(32)
here ts is the sampling period. This function is acoordinate-transformed version of the spatiotemporaltransfer function derived in Eq. ~31!, where the tiltedloci in the spatiotemporal frequency representationare mapped onto the orthogonal sampling of theAOA-temporal frequency representation.
In Fig. 10, beam forming is performed only with thedesired repetitive chirp in the input, without the jam-mers. Note the strong response in the direction ofthe desired signal ~0.25 rad!, over almost the entirechirp bandwidth ~0.5 to 2.5 GHz!, spatially blurred bythe angular resolvability of the finite rectangular ap-erture of the array, which of course varies with fre-quency. It is interesting to note that, even thoughthe chirp has a Gaussian window, the system trans-fer function is almost flat over the full bandwidth ofthe signal. This is explained by the fact that thesystem adapts to produce an output that resembles
the desired signal as closely as possible. Since boththe input and the desired signals are Gaussianchirps, it is to be expected that the system responseshould be flat over the entire signal bandwidth untilnoise or jammers start to dominate the signal. Alsonote that maximum receptivity describes an untiltedline at a constant unsquinted angle that is charac-teristic of a TTD system. However, the mainlobewidth does vary with frequency as do the positions ofthe sidelobes and nulls. This is to be expected, sinceadaptation in the presence of signals and antennanoise has optimized the response of the main lobeonly.
In Fig. 11, both the signals and the jammers arepresent in the signal environment, so beam formingand jammer nulling are performed simultaneously.Note the changes compared with Fig. 10. The mainlobe’s bandwidth has been slightly reduced, but itstill covers the full 1ye spectrum of the signal ~1.1 to1.9 GHz!. Part of this reduction is explained by thepresence of narrow-band jammer 2 at 0.8 GHz, on thefirst sidelobe of the main beam. Since the spatialsidelobe of this jammer extended onto the main beamand throughout its sidelobes, an improved error per-formance occurs by production of a null in that fre-quency, which covers a wide span of AOA’s withoutany detrimental effect on the received signal. Thewideband jammer 1 produced a long and thin null,located exactly at its AOA ~20.2 rads!, spanning itscomplete bandwidth ~from 0.5 to 2.5 GHz! withoutny squint of the null, and a width that is signifi-antly less than then main beam of the array. Bothammers were nulled to a depth of 45 dB with respecto the power of the detected signal of interest.
4. Conclusion
We have presented, analyzed, and simulated an ap-proach to broadband beam forming called BEAMTAP~broadband and efficient adaptive method for true-time-delay array processing! that reduces the num-er of TDL’s required for time-domain adaptive beamorming of a broadband N-element antenna from theonventional value of N TDL’s to only 2. For largerrays, this enables a dramatic hardware savingshat will allow for the implementation of broadbanddaptive phased arrays in sizes that were previouslympractical. The rf photonic implementation of thisEAMTAP algorithm is ideally matched to thetrengths of optical phased-array processing systems.e have presented an optical architecture based on
his new hardware-efficient adaptive algorithm thatses a fiber-remoted coherent phased array, a pho-orefractive crystal, AOD, and a synchronously scan-ing TFD, whose electronic output represents thedaptively beam-steered and jammer-nulled output.imulations were presented that verified this opera-ion for broadband desired signals in the presence ofarrow-band and broadband jammers.
We gratefully acknowledge the support and theirection of William Miceli of the Office of Naval Re-earch and the Office of the Secretary of Defense,
10 January 2000 y Vol. 39, No. 2 y APPLIED OPTICS 229
18. K. Wagner, S. Kraut, L. Griffiths, S. Weaver, R. T. Weverka,
3
2
Director of Defense Research and Engineering,through Multidisciplinary University Research Ini-tiative program grant N00014-97-1-1006; the insightof Anthony Sarto of Ball Aerospace; and our collabo-rations with Daniel Dolfi of Thomson-CSF. In addi-tion, we fondly remember the guidance andassistance by the late Brian Hendrickson of RomeLabs.
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